A charge Q located at a point `vec r` is in equilibrium under the combined field of three charges `q_(1), q_(2)` and `q_(3)`. If the charges `q_(1)` and `q_(2)` are located at point `vec r_(1)` and `vec r_(2)` respectively, find the direction of the force on Q due to `q_(3)` in terms of `q_(1),(q_2), vec r_(1) vec r_(2)`, and `vec(r)`.

Explanation

To find the direction of the force on charge \( Q \) due to charge \( q_3 \) in terms of \( q_1, q_2, \vec{r}_1, \vec{r}_2, \) and \( \vec{r} \), we can follow these steps:

### Step 1: Understand the Equilibrium Condition
Since charge \( Q \) is in equilibrium under the influence of the forces from charges \( q_1, q_2, \) and \( q_3 \), the net force acting on it must be zero. This can be expressed mathematically as:
\[ \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = 0 \]
where \( \vec{F}_1 \) is the force on \( Q \) due to \( q_1 \), \( \vec{F}_2 \) is the force on \( Q \) due to \( q_2 \), and \( \vec{F}_3 \) is the force on \( Q \) due to \( q_3 \).
...